Abstract
The optimal control problem for a shallow water equation with a viscous term is analyzed. The existence of optimal control to the control problem is investigated. The necessity condition of optimal control is derived by using the first order Gâteaux derivative of cost functional and adjoint equation. The local uniqueness of the optimal control is established by means of the second order Gâteaux derivative of cost functional. The novelty of this paper is that the necessity condition and local uniqueness of optimal control to the problem are obtained with viscous coefficient varepsilon>0.
Highlights
1 Introduction This paper is concerned with the optimal control problem for a shallow water equation with a viscous term, ut – uxxt + 2kux – ε(uxx – uxxxx) + muux = auxuxx + buuxxx, (1.1)
Motivated by the work in [6, 20, 29, 30, 32, 33], we studied the optimal control problem for the shallow water equation with a viscous term min J(v) =
The necessity condition of optimal control is derived by using the first order Gâteaux derivative of the cost functional and the adjoint equation
Summary
The optimal control and existence of optimal solution to the control problem are presented. Shen [11,12,13] studied the optimal control problem for a generalized viscous shallow water equation. Motivated by the work in [6, 20, 29, 30, 32, 33], we studied the optimal control problem for the shallow water equation with a viscous term min J(v) =
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